# Isentropic process

In thermodynamics, an isentropic process is a fictive idealized thermodynamic process that is adiabatic and in which the work transfers of the system are frictionless; there is no transfer of heat or of matter and the process is putatively reversible.[1][2][3][4]

## Background

The second law of thermodynamics states that,

$TdS \ge \delta Q$

where $\delta Q$ is the amount of energy the system gains by heating, $T$ is the temperature of the system, and $dS$ is the change in entropy. The equal sign will hold for a reversible process. For a reversible isentropic process, there is no transfer of heat energy and therefore the process is also adiabatic. For an irreversible process, the entropy will increase. Hence removal of heat from the system (cooling) is necessary to maintain a constant entropy for an irreversible process in order to make it isentropic. Thus an irreversible isentropic process is not adiabatic.

For reversible processes, an isentropic transformation is carried out by thermally "insulating" the system from its surroundings. Temperature is the thermodynamic conjugate variable to entropy, thus the conjugate process would be an isothermal process in which the system is thermally "connected" to a constant-temperature heat bath.

## Isentropic processes in thermodynamic systems

T-s (Entropy vs. Temperature) diagram of an isentropic process, which is a vertical line segment.

The entropy of a given mass does not change during a process that is internally reversible and adiabatic. A process during which the entropy remains constant is called an isentropic process, written $\bigtriangleup s=0$ or $s_1 = s_2$. [5] Some isentropic thermodynamic devices include: pumps, gas compressors, turbines, nozzles, and diffusers.

### Isentropic efficiencies of steady-flow devices in thermodynamic systems

Most steady-flow devices operate under adiabatic conditions, and the ideal process for these devices is the isentropic process.The parameter that describes how efficiently a device approximates a corresponding isentropic device is called isentropic or adiabatic efficiency.[6]

Isentropic efficiency of Turbines:

$\eta _T = \frac{\text{Actual Turbine Work}}{\text{Isentropic Turbine Work}}=\frac{W_a}{W_s} \cong \frac{h_1-h_{2a}}{h_1-h_{2s}}$

Isentropic efficiency of Compressors

$\eta _C = \frac{\text{Isentropic Compressor Work}}{\text{Actual Compressor Work}}=\frac{W_s}{W_a} \cong \frac{h_{2s}-h_1}{h_{2a}-h_1}$

Isentropic efficiency of Nozzles

$\eta _N = \frac{\text{Actual KE at Nozzle Exit}}{\text{Isentropic KE at Nozzle Exit}}=\frac{V_{2a}^2}{V_{2s}^2} \cong \frac{h_1-h_{2a}}{h_1-h_{2s}}$

For all the above equations:

$h_1$ is the enthalpy at the entrance state
$h_{2a}$ is the enthalpy at the exit state for the actual process
$h_{2s}$ is the enthalpy at the exit state for the isentropic process

### Isentropic devices in thermodynamic cycles

Ideal Rankine Cycle 1->2 Isentropic compression in a pump
Ideal Rankine Cycle 3->4 Isentropic expansion in a turbine
Ideal Carnot Cycle 2->3 Isentropic expansion
Ideal Carnot Cycle 4->1 Isentropic compression
Ideal Otto Cycle 1->2 Isentropic compression
Ideal Otto Cycle 3->4 Isentropic expansion
Ideal Diesel Cycle 1->2 Isentropic compression
Ideal Diesel Cycle 3->4 Isentropic expansion
Ideal Brayton Cycle 1->2 Isentropic compression in a compressor
Ideal Brayton Cycle 3->4 Isentropic expansion in a turbine
Ideal Vapor-compression refrigeration Cycle 1->2 Isentropic compression in a compressor
NOTE: The isentropic assumptions are only applicable with ideal cycles. Real world cycles have inherent losses due to inefficient compressors and turbines. The real world system are not truly isentropic but are rather idealized as isentropic for calculation purposes.

## Isentropic flow

In fluid dynamics, an isentropic flow is a fluid flow that is both adiabatic and reversible. That is, no heat is added to the flow, and no energy transformations occur due to friction or dissipative effects. For an isentropic flow of a perfect gas, several relations can be derived to define the pressure, density and temperature along a streamline.

Note that energy can be exchanged with the flow in an isentropic transformation, as long as it doesn't happen as heat exchange. An example of such an exchange would be an isentropic expansion or compression that entails work done on or by the flow.

### Derivation of the isentropic relations

For a closed system, the total change in energy of a system is the sum of the work done and the heat added,

$dU = \delta W + \delta Q\,\!$

The reversible work done on a system by changing the volume is,

$dW = -pdV\,\!$

where $p$ is the pressure and $V$ is the volume. The change in enthalpy ($H = U + pV\,\!$) is given by,

$dH = dU + pdV + Vdp\,\!$

Then for a process that is both reversible and adiabatic (i.e. no heat transfer occurs), $\delta Q_{rev} = 0\,\!$, and so $dS=\delta Q_{rev}/T = 0\,\!$. All reversible adiabatic processes are isentropic. This leads to two important observations,

$dU = \delta W + \delta Q = -pdV + 0\,\!$ , and
$dH = \delta W + \delta Q + pdV + Vdp = -pdV + 0 + pdV + Vdp = Vdp\,\!$

Next, a great deal can be computed for isentropic processes of an ideal gas. For any transformation of an ideal gas, it is always true that

$dU = nC_vdT\,\!$, and $dH = nC_pdT\,\!$.

Using the general results derived above for $dU$ and $dH$, then

$dU = nC_vdT = -pdV\,\!$, and
$dH = nC_pdT = Vdp\,\!$.

So for an ideal gas, the heat capacity ratio can be written as,

$\gamma = \frac{C_p}{C_V} = -\frac{dp/p}{dV/V}\,\!$

For an ideal gas $\gamma\,\!$ is constant. Hence on integrating the above equation, assuming a perfect gas, we get

$pV^{\gamma} = \mbox{constant} \,$ i.e.
$\frac{p_2}{p_1} = \left(\frac{V_1}{V_2}\right)^{\gamma}$

Using the equation of state for an ideal gas, $p V = n R T\,\!$,

$TV^{\gamma-1} = \mbox{constant} \,$
$\frac{p^{\gamma -1}}{T^{\gamma}} = \mbox{constant}$

also, for constant $C_p = C_v + R$ (per mole),

$\frac{V}{T} = \frac{nR}{p}$ and $p = \frac{nRT}{V}$
$S_2-S_1 = nC_p \ln\left(\frac{T_2}{T_1}\right) - nR\ln\left(\frac{p_2}{p_1}\right)$
$\frac{S_2-S_1}{n} = C_p \ln\left(\frac{T_2}{T_1}\right) - R\ln\left(\frac{T_2 V_1}{T_1 V_2}\right ) = C_v\ln\left(\frac{T_2}{T_1}\right)+ R \ln\left(\frac{V_2}{V_1}\right)$

Thus for isentropic processes with an ideal gas,

$T_2 = T_1\left(\frac{V_1}{V_2}\right)^{(R/C_v)}$ or $V_2 = V_1\left(\frac{T_1}{T_2}\right)^{(C_v/R)}$

### Table of isentropic relations for an ideal gas

 $\frac{T_2}{T_1}$ $=\,\!$ $\left (\frac{p_2}{p_1} \right )^\frac {\gamma-1}{\gamma}$ $=\,\!$ $\left (\frac{V_1}{V_2} \right )^{(\gamma-1)}$ $=\,\!$ $\left (\frac{\rho_2}{\rho_1} \right )^{(\gamma - 1)}$ $\left (\frac{T_2}{T_1} \right )^\frac {\gamma}{\gamma-1}$ $=\,\!$ $\frac {p_2} {p_1}$ $=\,\!$ $\left (\frac{V_1}{V_2} \right )^{\gamma}$ $=\,\!$ $\left (\frac{\rho_2}{\rho_1} \right )^{\gamma}$ $\left (\frac{T_1}{T_2} \right )^\frac {1}{\gamma-1}$ $=\,\!$ $\left (\frac{p_1}{p_2} \right )^\frac {1}{\gamma}$ $=\,\!$ $\frac{V_2}{V_1}$ $=\,\!$ $\frac{\rho_1}{\rho_2}$ $\left (\frac{T_2}{T_1} \right )^\frac {1}{\gamma-1}$ $=\,\!$ $\left (\frac{p_2}{p_1} \right )^\frac {1}{\gamma}$ $=\,\!$ $\frac{V_1}{V_2}$ $=\,\!$ $\frac{\rho_2}{\rho_1}$

Derived from:

$pV^{\gamma} = \text{constant} \,\!$
$pV = m R_s T \,\!$
$p = \rho R_s T\,\! \,\!$
Where:
• $p\,\!$ = Pressure
• $V\,\!$ = Volume
• $\gamma\,\!$ = Ratio of specific heats = $C_p/C_v\,\!$
• $T\,\!$ = Temperature
• $m\,\!$ = Mass
• $R_s\,\!$ = Gas constant for the specific gas = $R/M\,\!$
• $R\,\!$ = Universal gas constant
• $M\,\!$ = Molecular weight of the specific gas
• $\rho\,\!$ = Density
• $C_p\,\!$ = Specific heat at constant pressure
• $C_v\,\!$ = Specific heat at constant volume

## References

• Van Wylen, G.J. and Sonntag, R.E. (1965), Fundamentals of Classical Thermodynamics, John Wiley & Sons, Inc., New York. Library of Congress Catalog Card Number: 65-19470

### Notes

1. ^ Partington, J.R. (1949), An Advanced Treatise on Physical Chemistry., volume 1, Fundamental Principles. The Properties of Gases, London: Longmans, Green and Co., p. 122.
2. ^ Haase, R. (1971). Survey of Fundamental Laws, chapter 1 of Thermodynamics, pages 1–97 of volume 1, ed. W. Jost, of Physical Chemistry. An Advanced Treatise, ed. H. Eyring, D. Henderson, W. Jost, Academic Press, New York, lcn 73–117081, p. 71.
3. ^ Borgnakke, C., Sonntag., R.E. (2009). Fundamentals of Thermodynamics, seventh edition, Wiley, ISBN 978-0-470-04192-5, p. 310.
4. ^ Massey, B.S. (1970), Mechanics of Fluids, Section 12.2 (2nd edition) Van Nostrand Reinhold Company, London. Library of Congress Catalog Card Number: 67-25005, p. 19.
5. ^ Cengel, Yunus A., and Michaeul A. Boles. Thermodynamics: An Engineering Approach. 7th Edition ed. New York: Mcgraw-Hill, 2012. Print.
6. ^ Cengel, Yunus A., and Michaeul A. Boles. Thermodynamics: An Engineering Approach. 7th Edition ed. New York: Mcgraw-Hill, 2012. Print.